In group theory, a divisible group is an abelian group G such that for any positive integer n and any g in G, there exists y in G such that ny = g. One can show that G is divisible if and only if G is an injective object in the category of Z-modules (abelian groups).

Examples

• Q is divisible, as additive abelian group
• More generally, every vector space over Q has a divisible underlying group.
• Every quotient of a divisible group is divisible. Thus, Q/Z is divisible.
• The p-primary component of Q/Z which is isomorphic to the p-quasicyclic group is divisible.
• Every existentially closed group (in the model theoretic sense) is divisible.

Structure theorem of divisible groups

Let G be a divisible group. One can easily see that the torsion subgroup Tor(G) of G is divisible. Since a divisible group is an injective module, Tor(G) is a direct summand of G. So

.

As a quotient of a divisible group, G/Tor(G) is divisible. Moreover, it is torsion free. Thus, it is a vector space over Q and so there exists a set I such that

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The structure of the torsion subgroup is harder to determine, but one can show that for all prime numbers p there exists I_p such that

where (Tor(G))_p is the p-primary component of Tor(G).

Thus, if P is the set of prime numbers,

.